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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8640, base_ring=CyclotomicField(144))
M = H._module
chi = DirichletCharacter(H, M([72,27,64,36]))
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gp:[g,chi] = znchar(Mod(67, 8640))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8640.67");
| Modulus: | \(8640\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(8640\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(144\) |
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sage:chi.multiplicative_order()
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gp:charorder(g,chi)
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magma:Order(chi);
|
| Real: | no |
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sage:chi.multiplicative_order() <= 2
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gp:charorder(g,chi) <= 2
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magma:Order(chi) le 2;
|
| Primitive: | yes |
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sage:chi.is_primitive()
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gp:#znconreyconductor(g,chi)==1
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magma:IsPrimitive(chi);
|
| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
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\(\chi_{8640}(43,\cdot)\)
\(\chi_{8640}(67,\cdot)\)
\(\chi_{8640}(283,\cdot)\)
\(\chi_{8640}(547,\cdot)\)
\(\chi_{8640}(763,\cdot)\)
\(\chi_{8640}(787,\cdot)\)
\(\chi_{8640}(1003,\cdot)\)
\(\chi_{8640}(1267,\cdot)\)
\(\chi_{8640}(1483,\cdot)\)
\(\chi_{8640}(1507,\cdot)\)
\(\chi_{8640}(1723,\cdot)\)
\(\chi_{8640}(1987,\cdot)\)
\(\chi_{8640}(2203,\cdot)\)
\(\chi_{8640}(2227,\cdot)\)
\(\chi_{8640}(2443,\cdot)\)
\(\chi_{8640}(2707,\cdot)\)
\(\chi_{8640}(2923,\cdot)\)
\(\chi_{8640}(2947,\cdot)\)
\(\chi_{8640}(3163,\cdot)\)
\(\chi_{8640}(3427,\cdot)\)
\(\chi_{8640}(3643,\cdot)\)
\(\chi_{8640}(3667,\cdot)\)
\(\chi_{8640}(3883,\cdot)\)
\(\chi_{8640}(4147,\cdot)\)
\(\chi_{8640}(4363,\cdot)\)
\(\chi_{8640}(4387,\cdot)\)
\(\chi_{8640}(4603,\cdot)\)
\(\chi_{8640}(4867,\cdot)\)
\(\chi_{8640}(5083,\cdot)\)
\(\chi_{8640}(5107,\cdot)\)
...
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sage:chi.galois_orbit()
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gp:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
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magma:order := Order(chi);
{ chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
\((2431,3781,6401,3457)\) → \((-1,e\left(\frac{3}{16}\right),e\left(\frac{4}{9}\right),i)\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 8640 }(67, a) \) |
\(1\) | \(1\) | \(e\left(\frac{53}{72}\right)\) | \(e\left(\frac{31}{144}\right)\) | \(e\left(\frac{17}{144}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{55}{72}\right)\) | \(e\left(\frac{1}{144}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{13}{72}\right)\) |
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sage:chi(x)
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gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
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magma:chi(x)
